Like garlic, running water is more universally applicable than to just vampires. Running water is believed to be a barrier to the supernatural. The origins of this idea come from many different cultures.
The Greeks widely held that rivers such as the Styx served as the passageway from the land of the living to the underworld. Most Greek myths that have a hero traversing to the underworld (Odysseus, Hercules, Orpheus, etc.) have them traveling the Styx. The ferryman Charon takes spirits to the land of the dead via his boat on the Styx. So rivers, not unlike crossroads (but that’s another post) were often seen as powerful passages.
The Romans, who adopted most Greek traditions as their own, likewise felt power from rivers; not only did they have rituals regarding rivers, but they viewed many rivers as special demarcations, such as the Tiber and the Rubicon.
The Bible has its fair share of respect for rivers with the Tigris and Euphrates passing through Eden. The Jordan River was the site where John the Baptist baptized his followers and Jesus Christ.
In neighboring Egypt, Moses was cast upon the Nile in a story similar to that of Osiris who was shut into a box. He died, lived again, and is seen as the god of agriculture and the underworld, symbolized by the river Nile.
Hindus hold the Ganges and the Indus to be sacred rivers, while the Chinese look to the Yangtze. Celtic myths describe that many faerie creatures are unable to cross over rivers, and even the folktale by Washington Irving “The Legend of Sleepy Hollow” has Ichabod Crane in a mad dash to make it across a bridge over running water to reach safety. Clearly this is a culturally universal belief with rivers representing life, death, and boundaries.
As for the effects on the supernatural, there’s no known effect on them. It acts as a barrier that cannot be crossed, much like the threshold. This is either due to its sacred nature, or because it represents a supernatural boundary. Some even hold that running water prevents witches from being able to use their magic on people (Rasputin from Russian folklore was disposed of in a river because of this belief).
Sidenote: Most of the rivers mentioned are major rivers, so it could simply be a logistical problem instead of a supernatural one as these are clearly impassable without some kind of means such as a boat. However, the story of Sleepy Hollow uses words like brook and stream, which suggest that a specific volume of water is not necessary, but it still must be big enough to require a bridge to cross it. So don’t think that turning a hose onto a vampire will be enough to deter it.